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RD-A12@ 662 MODEL IDENTIFICATION AND ESTIMATION OF NONGAUSSIRN ARMA i/iPROCESSES(U) CALIFORNIA UNIV SRN DIEGO LA JOLLA DEPT OFMATHEMATICS K LII 82 N88@14-81-K-8883
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MICROCOPY RESOLUTION TEST CHARTNATIONAL BUREAU OF STANDAROS-1963-A
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-4
. . ,. . . . . . . . . - 4 . , . . . .. . . . . o.... A7 .. , .
0Model Identification and Estimation of NonGaussian ARMA Processes
- By
Keh-Shin Li
University of California, Riverside
Research supported In part by Office of Naval Research Contract100014-81-K-0003
Key words: ARMA model, Identification, Estimation, C-table, Pade
table, Asymptotic8, Bispectrum.
AMS 1970 subject classifications: Primary 62415; Secondary 62G05,62E70.
,TIG'K OCT 2 5 1982
jhdcr.tbgufOs82 10 22 004 AA!A
for publ.ic teloae cnd salet to
ditrlibutlom Is uzzlld.
I
Summary
Finite parameter model$of ARMA type have been used extensively in
many applications. Under the usual Gaussian assumption, the second
order analysis will not be able to discriminate among competing models
which give the same correlation structure. In many applications the
innovation process is non-Gaussian. In this case, analysis using
higher order moments will identify- the model uniquely without the
usual invertibility assumption. This in turn will affect the fore-
casting based on the non-Gaussian. model. We present a method which
uses bispectral analysis and the Pade approximation. We show that the
method will consistently identify the order of the ARMA model and
estimate the parameters of the model. One could also deconvolve the
process to estimate the innovation process ,,iich will provide infor-
mation for possible more efficient aximn likelihood estimation of
the parameters. Asymptotic distributions are given, and a few ex-
amples are presented to illustrate the effectiveness of the method.
Je.son For
VT q GRAkI
J;'tif lw't nr2
. . . ...- od..
iAvvil azid/or4, :6p c1l3
1. Introduction
Finite parameter autoregressive moving average models have been
used extensively in time series modeling, forecasting and control.
Host of the literature is concerned with Gaussian processes. Let ran-
don variables et, t - ...,-1,0,1,... be independent and identically
distributed with mean zero, tat M 0, and variance one Ee 2 - 1. Let
{aj } be a sequence of real constants with
Ea2<rEs <..
Consider the linear process generated by fa and let)
Xt Z a . (.I)lt j.-o3 et-i
The frequency response function is given by
A(e - m jeJ)= . (1.2)
If the process X is normally distributed then its full probabilityt
structure Is completely determined by its spectral density function
fiA) = I y _A 13
Hence the phase information In A(e - ) Is not Identifiable In the
Gaussian case. If A(z) Is a rational function
A(z) - Q (z)/P (s) (1.4)q p
with
q I
Qqlz) - I qz q 0 00
' (1.5)
.Pp(Z =rp~z PO0 -1
1-o
I
-2-
Nthen we say that the process (Xt satisfies a finite parameter auto-
regressive moving average model or simply ARhA(p,q). Usually we writeP (B)X t Q (B)e
(1.6)p t q t
where B is the backshift operator. There are two related problems to
be considered here. The first is to determine the orders of the poly-
nomials P (z) and Q (W). This Is the model identification problem.p q
The second is the problem of estimating the coefficients in P (z) andp
Q (z) after the model is identified. Given a model of the form (1.6),q
most of the literature assumes that P (z) and Q (W) have no root onp q
the unit disk, <~ <1. The condition P (z) * 0 for all I~ 1 is
called the realizability condition so that at has a one-sided infinite
*i moving average representation
X - A(B)e t - Ea e (1.7)tJ O i t-J
with
A(B) - E aJ-0
This is the same as saying a - 0 for all J < 0 in (1.1). The condi-
ti q Q 2<, z 1. called the Invertibility condition, is not
needed for stationarity. If Ix d satisfies (1.6) and Is a Gaussian
process then it Is well known that any real root rj * 0 of P () orp
Q" (z) can be replaced by its inverse r 1 and paired conjugate complex
roots can be replaced by their conjugated inverses rj without' chang-
Ing the correlation structure of (XtJ. This mans that if all the
roots are real and distinct then there are 2p + q different ways to
specify the roots and they are indistinguishable by examining the
autocorrelation function. Since different set of roots correspond to
q " ,' ", ,, .- , - ". '..- ": .. ' ." " ". '" '. . " "".. ." " ... .. ".. . ..
-3-
different set of coefficients, it is customary to assume that all
roots of Pp(z) and Qq( ) are outside the unit circle and to estimate
p q2.the coefficients of Pp(Z) and Q (s) under this condition. We will
present a method that can be applied without imposing the Invertibil-
ity assumption.
There are various procedures In the literature concerning the
identification of the orders p of P (z) and q of Q (z). Most of these
procedures involve the examination of the residuals or estimates of
6 tIs. In doing so, invertibility Is assumed. The distribution of t
is assumed to be Gaussian or a known one so that maximum likelihood
estimation of the coefficients can be carried out. Box and Jenkins
[1976] considered an Iterative procedure by examining the autocorrela-
tion function and partial autocorrelation function. In a series of
papers, Akaike [1969, 1971, 1978 proposed a final prediction error
criteria (FPF), an information criteria (AIC) and a Bayesian version
of it (BIC). These methods are based on multiple decision procedure
and were studied by others. (See Priestley [19811). Hypothesis
testing methods were considered by Godfrey [19791 and Poskitt and
Tremayne [19801. Gray, Kelly and Woodward 119781 considered the S
Array method using a pattern recognition technique. Nore recently
Woodward and Gray 119811 proposed a generalized partial autocorrela-
tion method. Tiso and Tasy [19811 proposed an Iterative regression
approach based on extended sutocorrelation function. Hannan and
Rlissanen [19821 considered a recursive method to identify an ARNAi model.
-4-
Parameter estimation methods have been developed by Hannan
[1969], Box and Jenkins [ 1976], Anderson 11977] and Ansley [1979).
If the process x tI is nonGaussian, LIi and Rosenblatt [19821
proved that, under broad conditions, (1.2) is Identifiable up to a
sign change and/or index shift of the aj'a requiring only that Pp (z)
and Q (z) have no root of absolute value one.q
It has been observed that in many geophysical and economic con-
text that data Is often nonGauselan. In this paper we propose a
method to Identify the orders p and q of the model (1.6) and estimate
the corresponding coefficients without the usual invertibillty assump-
tion. In section 2 we adopt the higher order spectrum method proposed
in Rosenblatt [19801 and Lii and Rosenblatt [19821 to estimate the
aj's In (1.7) and obtain their asymptotic distributions. In section
3 we Introduce C-table and the Pade table and give a method to
Identify the model and to estimate the underlying parameters. Asymp-
totic results are given. Section 4 consists of a few examples and a
discussion.
2. Asymptotics of the higher order spectral method
Let the frequency response function from (1.2) be
1
.e - 2f() explih()) (2.1)
There are many references concerning the estimation of the spectral
*. density function f(A). (See Anderson [19711 or Jenkins and Watts
119681). Lii and Rosenblatt 119821 proposed a method to estimate the
phase information h() when the process Xt (and hence the Innovation
process •t) is nonGaussian. Some basic results from this paper are
summarized in the following lemmas.
' " ".................... "" ......... . -" "'-,, .v, ,.. *. . ,-,
.1* -5-
Lama 2.1. Let {X I be a nonGausslan linear process given In (1.1)t
with the independent random variable {et ) having all moments finite.
Assuming that
IE IJa fand
,'. -i AA(e ) 0 for all I
andh(O) - 0 (2.2)
,! Then the phase h()) in (2.1) is given by
ho)) - hi(A) - Xhi(w)/, + aX (2.3)
where a Is an integer and
S1(1) - (b'u) - h'(0))du (2.4)
with
h'(0) - h'(A) - la 1 (2.5)
A.O
where aO2 is an integer such that C3, the ath order cumulant of (Xtj,
is nonzero and
h(o) +...+ h(_) - h( +...+ A 1)
A C -1 b(A1 ,...,9 _)1 (2.6)ag I AO)I a A-
! th
where b(.) Is the a order cumulant spectral density of the process
{Xtj discussed in BrIllinger and Rosenblatt (1967).
Remark 1. From (1.2) and (2.1) we have
A(.) - Ia. - ,2f(O)I exp.i,(o) •
Since a's are real, we have either Ia > 0 or Ea < 0. The assump-
tion h(O) - 0 in Lemms 2.1 represents an arbitrary choice of the signs
o 4 4 *~4* ~ . . . . . . . . .
L - . 4-
I,'. ':, .- ., " " ". .', ",' '.- .-. '..' -' --. -. ..- .-. . ..- --. -. . - .... -. . ..-. -.° • ,,;. .. . -- - - . .-.-... .. ..•.--. -.. -,- --
K -6-
of a 's. Observing Xt I only, the signs of the aj ' are-intrinsically
undecidable since we can mltiply all a 's and at's by minus one with-
out changing (1.1).
The integer a in (2.3) is Intrinsically undecidable also since it
corresponds to reindexing the Xf'a.Xt
Remark 2. However, in the usual normalization of model (1.6) or (1.7)
we assume so > 0. Under this assumption we can use Theorem 2.1 to be
proved later to ascertain the first nonzero ea and shift the Index and
adjust the sign accordingly. Without loss of generality, in what fol-
lows, we will assume, that u - 3 in order to illustrate the techniques
of the method.
Lema8 2.2. Under the assumptions of Lesa 2.1. An estimate of h1(X)
is, from (2.4 - 2.6),
k-IH(X)-- E arg b (jA,6) (2.7)n J-i-I
where kA - A and it is understood that the bispectral estiuates b (.)n
based on a sample of size n are weighted averages of third order
periodogram values. If b(X,u) e C2 and the weight function W Is
symetric and band limited with band width A, then
Hn(X) - hI(X) - %(X) + o p(H(X) -with
ERn( A) - AG( A) + o(A)eand Cov(R ()),R (U)) - f(O) Ortn( A, v) f2(u) du
W2(u,v)dudv (2.8)
2w 2 mn(AU) W2(u,v)dudv
a 3nC2
for A(n) 0 0, A2n * . as n + -
. - .. -.. - . . . . . . . .
*.... -- - - -. . . . . . ---.. . . ......-. ........... .. .. ,... .-- .. "°.. . .. ". o.. . -
-7-
where G ) is a function involving b(,p). Further ER (A) + h I(I)
and H (A )'s are asymptotically jointly normally distributed with co-n j
variances given by (2.8).
Since
1 2w A~e ix)eJ: ai f- Me )e dX
an estimate as of a Is given byA, ^1 2 w MA
aj 2 I A~-l)et~ dl
1H an(T)-- E (2wfn(Xk)k7 exp{i(H n(X) -' + J") (2.9)
k-i I l
which by symetry can be written as
I2 E (2fnk-i 1n~ -
where 2L - M - 2w/A and "k represent a discretization and f is anIt n
estimate of f(.) similar to that of b(.). For a given sample of size
n, let the bandwidth of the weight function WI in fn(X) be A, and the
bandwidth of the weight function W2 in b(Xp) be A2, we will derive
the asymptotic joint distribution of the aj 's given in (2.9) to the
first order.
It is proved in Brillinger and Rosenblatt (1967) that if for
1-1,2 A + 0 and nA1 . " as n + W, then asymptotically as n + -
fn (X) and ba(X jPi) are independent normally distributed with
var(fnAk)) . f 21) W2 u)du O .
Since Hn is a function of bn, hence Hn and f are asymptotically inde-e nde n
pendent. Let
;,dA f2 (A)Cos(Zn(A )+kX)
withH (W)
Zn(A) - Hn(l) - 2 A
and observe that the asymptotic distribution of the vector (f n(AI)
H l(A), f (X ), H ( ) (w)) is normal with mean (f( h(X).aI n j n j nf(X ), h (A j) h I(w)) and covariance matrix
a A 0 a 0 0
' 0 t 0 t I t T
5 0 a 0 0
0 ta ,l 0 tj ,j tj ,W
0 t 0 t to t 0,t t~
L, i. jW .V
A where
8 s£j Cov(fn(A 1) If(j))
t Cov(H(),Hn( )) with A w
we note that the magnitude of st,1 Is smaller than that of t, . An
application of a multivariate 6-method (see Bishop, Fienberg and
Holland [1975] p. 493) we can show that the asymptotic distribution of
(d Ik dj,m ) is bivarlate normal with mean
1 1
(f 2( AX,)CoS(Z( 1A)+klX) ,f 2 ( A )Cos(Z( A1 )4UA ))
* whereZ(Ax) = hi(x) - Ath (T)/W
and coveriance matrix'.
A1
-9-
C(l,k;L ,k) C(tk;jm)
with
C(l,k;j,) f 2 (A)f 2 (A) Sin(Z(Xt)+kXt)
ndSin(Z(XA )ftm A) (min(A., A~ - Xtjw (.0and
22 2X W 3 2w 2 W(u,v)dudv
Using this and a straightforward calculation, we have the following
theorem.
Theorem 2.1. Under the assumption of Lemma 2.1 w have ( - ak) for
k-I,...,K are asymptotically jointly Gaussian with mans zero and co-
variances given by
? Cv( k~a ) -2(2w)3 fW2(u~vdd L LCov(a, ) - vdu X E C(l,k;j,m) (2.11)
where C(lk;j,m) is given in (2.10)
We will now assume that the stationary process Ix d satisfies
(1.6) with a representation given in (1.7) such that a0 > 0. As
usual, we assume P p(z) and Qq (z) given In (1.5) have no common factor
with P0 - I and q0 > 0. Under these assumptions, equation (2.11) canbe used to estimate the variance of a with f(A) and h (A) estimated
by fa(X) and Hn(X I ) respectively. These results can be used to deter-
mine the smallest integer k such that k * 0. We then reindex the
aj's and change their signs if necessary. This gives a complete pro-
cedure to estimate a 's in equation (1.7) consistently. We use these
estimated A ts to identify and estimate the polynomials P (z) anda pQ (z) by the C-table and the Pads tc .I- as discussed in Lii [19821q
-10-
dealing with a distributed lag model.
3. Asymptotics of the C-table and the Pade approximant
Given a pair of nonnegative integers q and p we denote Pade
rational approximants to a formal power series A(z) - £ a zj by1-0 J
[q/pj - Qq (z)/Pp (z) where Qq (z) and Pp(z) are polynomials of degrees
at most q and p respectively. We assume P (0) - 1 and Q (z) and P (z)P q p
have no common factors. The coefficients of Q (z) and P (z) are
determined by A(z) - (Q )(z)/F ()) - o(z P+q+). The following three
lemmas can be found in Baker [19751.
Lemma 3.1. When it exists, the (q/p] Pade approximant for A(z) is
uniquely determined. Further
a a too aq-p+l q-p+2 q 1
aq-p+2 aq-p+3 a q+2
Qq(z) det a a ... a (3.1)q q q+1 q+p(31
q q qEa z Z E a z ... q a.
jMp i-P J-p-I -ar J=O
and
aq-~l aq-p42 too aq+1
aq-p+2 aq-p+3 q+2
Pp(z) det (3.2)
aq aq+1 aq+p
zp z p - I
where a = 0 if j < 0 and the summation is set to zero if the lower
index on a sum exceeds the upper index.
i
i? -11-
Given nonnegative integers r and s, we define
ar-s+1 ar-s+2 •. ar
C det a a (3.3)• :r,s ar-s+2 ar-s+3 "" r+1
r r+1 r+8-i
5(5-1)
- (-1) = det[(ari.j)i,j a 11
The C-table,which is a doubly infinite array, is defined by
C - (C ) . We further define C -1.rs rs rO
Lemma 3.2. (i) C * 0 implies that [q/p] exists. (ii) Every zeroqP
a
entry in the C-table for a formal power series A(z) - I + E ajzJ-1
occurs in a square block of zero entries and Is completely boarded by
nonzero entries.
Lemma 3.3. Given a formal power series A(z) the following three
conditions are equivalent
£ m
(1) A(z). E c zJ/(1 + E d zj)
(2) [q/pJ A(z) for all q > I and p m
(3) C 0 and C - 0 for all r > I and s > m.
qp r,s
If condition (3) in Lemna 3.3 is satisfied, we call the entry
(1+1, m+1) in the C-table the "breaking point".
Lemmas 3.1, 3.2 and 3.3 lead to the following.
Theorem 3.1. The process JXt given in (1.7) is ARMA (p,q) given in
(1.6) if and only if the C-table associated with A(z) has the breakingI
point (p+l,q+l). Further C * 0 and the coefficients of P (z) and•q ,p p
Q (z) in (1.6) are obtained from (3.1) and (3.2). To normalize theseq
.
-12-
coefficients we divided both P (z) and Qq (z) by C qp so that P- 1.
Whether the roots of Q (z) are inside or outside of the unit circle isq
Immaterial here.
This theorem provides a consistent procedure to identify the
model by determining the orders p and q and to estimate the parameters
of the identified model. We use the a 's obtained from (2.9) to con-
struct estimates Cr,s of C r,s, ascertain the breaking point in
C-table to identify the model and finally by substituting the a 's
for ai's in (3.1) and (3.2) to obtain estimates of the coefficients of
F P(z) and Qq (z). The following lemmas can now be proved with simple
modifications of the proofs given in section 4 of Lit [1982].Lemma 3.4. If Ii.aJKj are asymptotically jointly Gaussian for a fixed
integer K with mean {aj jK.! and covariance matrix X(Cn) where2Cn n 0 6 > 0 as the sample size n then the asymptotic dis-
tribution of R, the determinant of an M*1 matrix,
R - det (aJ-)IjJ with a a ia~1, Is Gaussian with mean
R - det [(a
I4
and variance
O2 . )Gt where Ct is the transpose of C - ( ...,gK) with
g R.4' i j a.
We note that computationally g1 is the sm of the cofactors of ea in
the matrix .[(a1 ,1 ,j..
[HF .
-13-
Theorem 3.2. If the estimates of faJ in (1.7) are given by fajI
obtained from (2.9), then for fixed r and a,
r,s r+i-j ij-1
is asymptotically normally distributed with mean C given In (3.3)
and variance GEG' where C - (g , ) with L - r-s+1,
U - r+s-l, gj = C a nd I is the covariance matrix of ( .Let'a'U)
from (2.11).
This theorem gives a method to construct the C-table and to find
the breaking point (q+l, p+1). If the breaking point can not be
- uniquely determined, the C-table will reduce the number of possible
competing models to only a few for further testing. If the process
IXtI does not have a rational frequency response function [q/p], the
C-table will still suggest a possible ARMA (pq) approximation to A(z)
using the principle of parsimony. Once we have, identified the model
to be ARMA (pq), replacing the a 's by their estimates a 'a in equa-
tions (3.2), we obtain estimates P^ and q of Pa and q respectively
by
Qq(z) q!4+1 lz + ... + 1-A q
Pp(-) j 14z + ... + At P
-" O z + ... + q q (3.4)
1+ ^ Ap p
with* 0
A(3.5)qi I ; I OVl,...,q *
To ottalsn the asymptotic distributions of p 's and ^q's we evaluate
P •
-14-
the determinant In (3.2) by cofactor expansion of the last row. (zP sP-1
( , • , 1) and w obtain
i *z _ 1 "~z 2 (-1)p A z (3.6)p
where I is the cofactor of ZI in (3.2).
Similarly, w have from (3.1)
-q a._pJ),. J.0 i - - pi_
Sa 0Ao + (a1 -aOA) +...+ +...+ (-) PA )qq-p
"BO + it3 + + B3z (3.7)
0 1 q
Using em 3.4 and equations (3.4) - (3.7) we can prove
Theorem 3.3. For fixed p and q, let L - maxfO,q-p+l and U - q+p-1.
Then the asymptotic distributions of ( j-Pj. -P ) - and
(~qj q-ql) are, each bivariate normal with mean (0.0) and covariance
matrices
':, E~PP P ,,ii .j~~ "(Ci.L.U) Z.LU(P.,-U. L~U).. ,LU
d Oi: . LU) L,U( L t t
zij~~. PLU 1,L,Uj q
where''~41**'U~i0,.,
OJ ..U (hL AhL+ 1 $**h U ) J-Op99,q
with oo , o
9 A -, qht B I -L,...,U
and
ZL,U - CoV( L ,P.", ) from (2.11).
A I and Bt are the theoretical values of A and BI respectively in
(3.6) and (3.7). Furthermore, the asymptotic distribution of pI
.and -q are normal with mean zero and variancesi J
1
P" P6 - )0 o ? )
4. Snple andDiscpslo2 q q fo
j 0 3he p o qj 2)P6J
-6
1.
h4. Examples and Discussion
Examples In this section are simulated according to the model of
the form P p(e)X - %q'B*e with
andP (S) * 0 when 1:< I
* The Innovation process are obtained from
* t a t
-16-
where e' are independent, identical, exponentially distributed witht
Sa' - 1 and 2 Var(e') - 1. Hence Et - 0, Var(e t ) - 1. The
sample size for X is 640. Some computational details are discussedt
in Lii and Helland 119811 and Lii and Rosenblatt [19821.
Example 1.
Q (o) - 1 - 0.6B + 0
PI(3) -.1 + 0.6B
AU the roots are outside of unit circle. Table I gives the C-table
associated with this model. Each entry has two numbers, the upper one
is C and the lower one is the estimated standard deviation of Crs rts
computed from Theorem 3.2. We also exhibit table 2 which gives the
* ratio of the C and its estimated standard deviation of each entryrps
in table 1. We call table 2 the "resolution table" of table 1. It is
such easier to recognize the pattern in a resolution table when there
is a sudden drop of resolution at entry (tm) and thereafter (I,=) is
;. likely to be the breaking point. From table 2, it is clear that (3,2)
- is the breaking point and the model is correctly identified as ARMA
(2,1). The Pads approximant 12/11 gives, from (3.4) and (3.5),
Q2 (B) - 1.068 - 0.585 1 + 0.763 3(0.446) (0.212) (0.097)
- andPI(B) - I + 0.594 B
(0.092)
where the numbers In the parentheses are estimated standard deviations
from theorem 103.
Example 2. In this example both roots, -0.5 and -0.75, of
02(B) - I + 3.53 + 332
",~~~~~~~~~~~~~~~~~~~~~. --. , -.. . . ....... . ..', ' . . .- -. .-'" -... _- _ . -. .-. ".•...-i.
-17-
* are inside of the unit circle while the roots of P2 (B) - 1 + 0.35B +
0.5B are outside of unit circle. The associated a-table Is table 3
and Its resolution table Is table 4. It seems reasonable to identify
the model to be ARIA(2,2) with breaking point at (3,3). The Pade
approximant [2/21 gives
Q2(B) = 0.907 + 3.64 B + 2.86 B2(0.94) (13.7) (10.5)
and P2(B) I 1 + 0.55 B + 0.53 B2 (4.1)
(0.60) (0.62)
The large estimated standard deviations in Example 2 may be due
to the complicated formula In Theorem 3.3 and the number of parameters
are large relative to the sample size. Nevertheless, the estimates of
the parameters provide good starting values for possible more effl-
cient Iterative methods. We note that the usual Iterative type of
fitting procedure can be used here. We can deconvolve the process X
ttSand estimate the Innovation process e t by s^t". Diagnostic checking can
be performed on et to discriminate among possible competing models.
The probability distribution or density function of e can be esti-
mated to facilitate a non-Gaussian maxima likelihood estimation. It
seems that in building a finite parameter ARMA model of a stationary
tim series {X t, one should use the procedure suggested In Lii and
Rosenblatt [19821 to deconvolve Xt and see if jetj is near Gaussian or
not. If not, one should use the procedure suggested in this paper to
build the AIMA model without Imposing the invertibility condition.
Alternatively, one may want to use any one of those methods mentioned
in the introduction section, using mainly the second order structure,
to identify the orders of the model; however one should still use the
Pade approximant to estimate necessary coefficients and to identify
p.-.v................................................ ,
77-7-18-
whether the roots lie inside or outside the unit circle. Even in the
Gaussian case, one may want to first fit an HA(K) for a moderate inte-
ger K (say 15). Then following the procedure in section 3, one can
identify the equivalent parsimonious ARMA(p,q) model and obtain estl-
mates of parameters. As a comparison, we employed the usual Box-
Jenkins type estimation procedure as it is implemented in the sub-
routine FTHL of the International Mathematical and Statistical LAbrary
(ISL). Given the right orders in the model, we obtain estimates
Q2(B) - 1.0 + 1.1611 + 0.302752
and (4.2)
?2 (B) - 1.0 + 0.3307B + 0.478932
. with estimated white noise variance o2 - 8.765.
Using Q2(3) in (4.2) to Interprete the model may be quite
different from that of using Q2(3) in (4.1).
Example 2 show that e can discriminate models which are Indis-
tinguishable using only second order properties. The method proposal
in this paper produce estimates that are consistent. For moderate
sample size this method can be a valuable tool for AlMA model identi-
fication and estimation.
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-23-
References
1. Akalke, H. (1969). "Fitting autoregressive model for predic-tion." Annals of the Institute of Statistical Mathematics21, 243-247.
2. Akaike, H. (1973). "Information theory and an extension of themaximum likelihood principle". In "2nd InternationalSymposium on Information Theory", Ed. by B. N. Petrov andF. Caski, 267-281, Akademia Kiado, Budapest.
3. Akalke, H. (1978). "A Bayesian analysis of the minimum AICprocedure." Annals of the Institute of Statistical Mathe-matics 30, 9-14.
4. Anderson, T. W. (1971). "The spectral analysis of time series."John Wiley, New York.
5. Anderson, T. W. (1977). "Estimation for autoregressive movingaverage models in the time and frequency domains." TheAnnals of Statistics 6, 842-865.
6. Ansley, G. F. (1979). "An algorithm for the exact likelihood ofa mixed autoregressive moving average process." Biometrica66, 59-65.
7. Baker, G. A., Jr. (1975). "Essentials of Pade approximations."Academic Press, New York.
8. Bishop, Y. M. M., Fienberg, S. E. and Holland, P. W. (1975)."Discrete miltivariate analysis - theory and practice." TheMIT Press, Cambridge, Massachusetts.
9. Box, G. E. P., and Jenkins, G. H. (1976). "Time series analysis:forecasting and control." Holden-Day, San Francisco.
10. Brillinger, D. R. and Rosenblatt, M. (1967). "Asymptotic theoryof estimates of kth order spectra." In "Spectral Analysisof Time Series", Ed. by B. Harris, 153-188. John Wiley, NewYork.
11. Chatfield, C. (1979). "Inverse autocorrelation." Journal of the
Royal Statistical Society A 142, 363-377.
Fl 12. Godfrey, L. G. (1979). "Testing the adequacy of a time seriesmodel." Biometrica 66, 67-72.K 13. Gray, H. L., Kelley, G. D., and Mclntire, D. D. (1978). "A newapproach to ARMA modeling." Communications in StatisticsB7, 1-77.
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14. Hannan, E. J. (1969). "The estimation of mixed moving averageautoregressive system." Biometrica 56, 579-594.
15. Hannan, E. J. and Rissanen, J. (1982). "Recursive estimation ofmixed autoregressive-moving average order." Biometrica 1,81-94.
16. Jenkins, G. M. and Watts, D. G. (1968). "Spectral analysis andits application." Holden-Day, San Francisco.
17. Lii, K. S. and Helland, H. N. (1981). "Cross-bispectrum computa-tation and variance estimation." ACM Transaction of Mathe-matical Software 7, 284-294.
18. Lii, K. S. and Rosenblatt, H. (1982). "Deconvolution and estima-tion of transfer function phase and coefficients for non-Gaussian linear processes." To appear in The Annals ofStatistics, December.
19. Lii, K. S. (1982). "Model identification in a transfer functionmodel." Technical report No. 95, Statistics Department,University of California, Riverside.
20. Poskitt, D. S. and Tremayne, A. R. (1980). "Testing the specifi-cation of a fitted autoregressive-moving average model."Biometrica 67, 359-363.
21. Priestley, M. B. (1981). "Spectral analysis and time series,Vol. 1." Academic Press, New York.
22. Rosenblatt, M. (1980). "Linear processes and Bispectra."Journal of Applied Probability 17, 265-270.
23. Tiao, G. C., and Tsay, R. S. (1981). "Identification of non-stationary and stationary ARMA models." In Proceedings ofthe Business and Economic Statistics Section. American Sta-tistical Association, Washington, D.C.
24. Woodward, W. A., and Gray, H. L. (1981). "On the relationshipbetween the S array and the Box-Jenkins method of ARMA modelidentification." Journal of the American Statistical Assoc-iation 76, 579-587.
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4. TITL.E (and Subtitle) S. TYPE OF REPORT & PERIOD COVERED
MODEL IDENTIFICATION AND ESTIMATION OF Research
NONGAUSSIAN ARMA PROCESSES 6. PERFORMING ORG. REPORT NUMBER
7. AUThOR(e) S. CONTRACT OR GRANT NUMBER(s)
Keh-Shin Lii ONR N00014-81-K0003
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19. KEY WORDS (Continue on reverse side If neccesary and Identify by block number)
ARMA model, identification, estimation, C-table, Pade table, Asymptotics,
bispectrum
20. ABSTRACT (Continue on reverse aide If necesary and Identify by block number)
Finite parameter models of ARMA type have been used extensively in many applica-tions. Under the usual Gaussian assumption, the second order analysis will notbe able to discriminate among competing models which give the same correlationstructure. In many applications the innovation process is nonGaussian. In thiscase, analysis using higher order moments will identify the models uniquelywithout the usual invertibility assumption. This in turn will affect the fore-casting based on the nonGaussian model. We present a method which uses bi-spectral analysis and the Pade avvroximi.tion. We show that the (continued)
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20. ABSTRACT (Continued)
method will consistently identify the order of the ARMA model and estimate theparamet ers of the model. One could also deconvolve the process to estimatethe innovation process which will provide information for possible more effi-cient maximum likelihood estimation of the parameters. Asymptotic distributionsare given, and a few examples are presented to illustrate the effectiveness ofthe method.
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